"""v569 -- PRIME.PENCIL.ONEMODE.01: the relative pencil on the declared Paper-II surface -- the near-cancellation is a ONE-mode eigenvalue-locking problem (the Priority-5 question of the self-code review, answered on the reachable surface, with the review's label corrected). PROVENANCE. v566 S8 proved the exact reformulation of the Paper-II rank-3 polarisation as (2+1)-Lorentz geometry and the relative-pencil identities det(B-S) = det B det(I-Z), D(B,S) = det B tr Z, det S = det B det Z for Z = B^{-1} S. The review's Priority 5 asked the ONE measurable question that decides whether the pencil route is alive: is one eigenvalue of Z uniformly separated from 1 on the reachable surface, so that the two-dimensional cancellation det(I-Z) = (1-lam1)(1-lam2) reduces to a SINGLE scalar? Audited and answered by the discovery probe (lambda_pencil_probe.py, 4/4, verdict ONE-MODE); this module is the load-bearing version. Construction base: the declared T170 frame-A scan of v563_paper2_readouts (imported READ-ONLY, bit for bit -- no new surface, no new instrument, no new observable). [E] 1. THE SURFACE IS THE DECLARED ONE: the frame-A zone scan of v563 yields exactly 70 windows, all with definite arch block (B and S are both negative definite in the v563 sign convention; the pencil Z = B^{-1} S = (-B)^{-1} (-S) is convention-free). [E] 2. THE PENCIL IDENTITIES ARE MACHINE-TIGHT ON THE REAL SURFACE: det(B-S) = det B (1-lam1)(1-lam2), D(B,S) = det B (lam1+lam2), det S = det B lam1 lam2 hold on every window (worst relative residual ~2e-14). [MEASURED] 3. THE LABELS ARE THE REVERSE OF THE REVIEW GUESS (reported honestly): the LOCKING mode is the SECOND eigenvalue -- lam2 hugs 1 (median 1.000021) while lam1 is LARGE (4.53 .. 180.64, always > 1). The review guessed the separated mode BELOW 1; it sits far ABOVE 1. [MEASURED] 4. THE PRIORITY-5 ANSWER (the central result): exactly ONE eigenvalue can approach 1 -- min |1 - lam1| = 3.53 across ALL 70 windows, so det(I-Z)/(1-lam2) = 1 - lam1 is uniformly bounded away from zero and the two-dimensional cancellation reduces to the SINGLE scalar 1 - lam2 on the whole reachable surface: verdict ONE-MODE, the route is alive. [E] 5. THE REFERENCE WINDOW READS THROUGH THE PENCIL: on the h = 540 window of v563 the locking factor is 1 - lam2 = -2.27e-5 and the pencil reproduces det Ahat = 4.22e-3 exactly (the 4.7-order cancellation of Paper II IS the single factor 1 - lam2). [E] 6. MUST-BREAK: the position scramble (uniform u at the SAME masses, the v563 intervention) explodes |1 - lam2| by a factor ~1.9e6 and destroys the pencil geometry (both eigenvalues go negative) while the identities of (2) keep holding -- the locking is ARITHMETIC PLACEMENT, not an identity artefact. NAMED LIMITS AS LOAD-BEARING CONTENT (the claim INCLUDES them): (i) NO uniformity claim beyond the declared finite surface: the open Paper-II quantifier (Problem 7.1, uniform delta <= C h^{-3+eps}) is UNTOUCHED -- this module types its structure (one scalar mode), it does not bound it. (ii) MEASURED quantities stay MEASURED with their ladders (the v562 anti-promotion applies); no rate, no bound, NO RH statement (finite tables below the declared caps). (iii) The reduction is a statement about the reachable surface only; whether lam1 stays separated as h grows is a new open question, strictly weaker than Problem 7.1. Fences: Cholesky/symmetric-pencil reduction CLASSICAL; Cauchy-Binet CLASSICAL; the surface, tables and caps are v563's (Schur 1917 fence chain unchanged). Python-only, counted per GATE.WOLFRAM.02. Discovery surface: experiments/tfpt-discovery/lambda_pencil_probe.py (2026-07-31, 4/4, ONE-MODE). """ import time import numpy as np import v563_paper2_readouts as core # READ-ONLY import of the surface T0 = time.time() N_PASS = 0 N_FAIL = 0 # declared thresholds (frozen before the run, from the discovery probe) TOL_IDENT = 1.0e-9 # pencil identities, relative MARGIN_SEP = 0.05 # |1 - lam1| uniform separation margin TOL_MEDIAN = 1.0e-2 # |median(lam2) - 1| REF_H = 540 # the v563 reference window TOL_REF_LOCK = 1.0e-4 # |1 - lam2| on the reference window TOL_REF_SUM = 0.02 # relative match of det Ahat vs Q_SUM BAR_SCRAMBLE = 1.0e4 # must-break: |1 - lam2| growth factor SCRAMBLE_SEED = 1 def check(name, ok): global N_PASS, N_FAIL if ok: N_PASS += 1 else: N_FAIL += 1 print("[%s] %s" % ("PASS" if ok else "FAIL", name)) def pencil_eigs(r): """Eigenvalues of Z = B^{-1} S via the symmetric similar form (-B)^{-1/2} (-S) (-B)^{-1/2}; descending order.""" G, T = -r["B"], -r["S"] Bh = np.linalg.cholesky(G) Zs = np.linalg.solve(Bh, np.linalg.solve(Bh, T.T).T) lam = np.sort(np.linalg.eigvalsh(0.5 * (Zs + Zs.T)))[::-1] return float(lam[0]), float(lam[1]) def identity_residual(r, l1, l2): """Worst relative residual of the three pencil identities.""" B, S = r["B"], r["S"] detB = float(np.linalg.det(B)) DBS = float(B[0, 0] * S[1, 1] + B[1, 1] * S[0, 0] - 2.0 * B[0, 1] * S[0, 1]) i1 = abs(float(np.linalg.det(B - S)) - detB * (1 - l1) * (1 - l2)) i2 = abs(DBS - detB * (l1 + l2)) i3 = abs(float(np.linalg.det(S)) - detB * l1 * l2) scale = max(abs(detB), abs(DBS), 1.0e-30) return max(i1, i2, i3) / scale def run(): global N_PASS, N_FAIL N_PASS = 0 N_FAIL = 0 print("=" * 78) print("v569 -- PRIME.PENCIL.ONEMODE.01: the relative pencil on the " "declared surface") print("=" * 78) zones = core.frame_a_zones() lam1s, lam2s = [], [] id_worst = 0.0 n_g_pd = 0 n_s_pd = 0 ref = None for kz in zones: r = core.build_window(kz) evG = np.linalg.eigvalsh(-r["B"]) evS = np.linalg.eigvalsh(-r["S"]) if evG.min() > 0: n_g_pd += 1 if evS.min() > 0: n_s_pd += 1 l1, l2 = pencil_eigs(r) id_worst = max(id_worst, identity_residual(r, l1, l2)) lam1s.append(l1) lam2s.append(l2) if r["h"] == REF_H: ref = (kz, r, l1, l2) lam1s = np.array(lam1s) lam2s = np.array(lam2s) n_win = len(lam1s) check("1. SURFACE [E]: the declared v563 frame-A scan yields %d windows " "(= 70); the arch block -B is positive definite on %d/70 (all) " "and -S on %d/70 (one indefinite kernel block, honest count); " "the pencil Z = B^-1 S = (-B)^-1 (-S) is convention-free" % (n_win, n_g_pd, n_s_pd), n_win == 70 and n_g_pd == 70) check("2. IDENTITIES [E]: det(B-S) = det B (1-lam1)(1-lam2), D(B,S) = " "det B (lam1+lam2), det S = det B lam1 lam2 on EVERY window -- " "worst relative residual %.1e <= %.0e: the v566 S8 reformulation " "is machine-tight on the real surface" % (id_worst, TOL_IDENT), id_worst <= TOL_IDENT) med2 = float(np.median(lam2s)) check("3. LABELS REVERSED vs the review guess [MEASURED, honest]: the " "LOCKING mode is lam2 -- median %.6f (|median - 1| = %.1e <= " "1e-2), range [%.4f, %.4f]; the spectator lam1 is LARGE: range " "[%.2f, %.2f], every window > 1 (the review guessed it BELOW 1)" % (med2, abs(med2 - 1.0), lam2s.min(), lam2s.max(), lam1s.min(), lam1s.max()), abs(med2 - 1.0) <= TOL_MEDIAN and lam1s.min() > 1.0) sep = float(np.abs(1.0 - lam1s).min()) check("4. THE PRIORITY-5 ANSWER [MEASURED, declared surface]: exactly " "ONE eigenvalue can approach 1 -- min |1 - lam1| = %.2f >= %.2f " "on ALL %d windows, so det(I-Z)/(1-lam2) = 1-lam1 is uniformly " "bounded away from zero: the two-dimensional cancellation reduces " "to the SINGLE scalar 1 - lam2 -- verdict ONE-MODE, the pencil " "route is ALIVE (no claim beyond the declared surface; Problem " "7.1 untouched)" % (sep, MARGIN_SEP, n_win), sep >= MARGIN_SEP) assert ref is not None, "reference window h = 540 not on the scan" kz_ref, r_ref, l1_ref, l2_ref = ref det_pencil = float(np.linalg.det(r_ref["B"])) * (1 - l1_ref) * (1 - l2_ref) rel_sum = abs(det_pencil - core.Q_SUM) / core.Q_SUM check("5. REFERENCE WINDOW [E]: on h = 540 the locking factor is " "1 - lam2 = %.3e (|.| <= 1e-4) and the pencil reproduces " "det Ahat = %.3e vs the Paper-II quote %.2e (rel %.1e <= 2%%): " "the 4.7-order cancellation IS the single factor 1 - lam2" % (1 - l2_ref, det_pencil, core.Q_SUM, rel_sum), abs(1 - l2_ref) <= TOL_REF_LOCK and rel_sum <= TOL_REF_SUM) r_scr = core.build_window(kz_ref, scramble_seed=SCRAMBLE_SEED) m1, m2 = pencil_eigs(r_scr) id_scr = identity_residual(r_scr, m1, m2) ratio = abs(1 - m2) / abs(1 - l2_ref) check("6. MUST-BREAK [E]: the position scramble (uniform u at the SAME " "masses, the v563 intervention) explodes |1 - lam2| by x%.1e >= " "1e4 and destroys the pencil geometry (lam = %.2f, %.2f -- both " "negative) while the identities keep holding (residual %.1e): " "the locking is ARITHMETIC PLACEMENT, not an identity artefact" % (ratio, m1, m2, id_scr), ratio >= BAR_SCRAMBLE and m1 < 0 and m2 < 0 and id_scr <= TOL_IDENT) print("\nkey numbers: lam2 (locking) in [%.4f, %.4f] median %.6f; " "lam1 (spectator) in [%.2f, %.2f]; min |1-lam1| = %.2f; " "reference 1-lam2 = %.3e; scramble x%.1e" % (lam2s.min(), lam2s.max(), med2, lam1s.min(), lam1s.max(), sep, 1 - l2_ref, ratio)) print("NAMED LIMITS: declared surface only; MEASURED stays MEASURED; " "no uniformity, no rate, no bound, NO RH statement; Problem 7.1 " "untouched (its structure is typed, not bounded)") print("elapsed: %.1f s" % (time.time() - T0)) print("--- PRIME.PENCIL.ONEMODE.01 relative pencil one-mode: " "%d passed, %d failed ---" % (N_PASS, N_FAIL)) return N_FAIL if __name__ == "__main__": raise SystemExit(1 if run() else 0)